Error Analysis of a High Order Method for Time-Fractional Diffusion Equations

An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data

“…The L1 scheme may be obtained by the direct approximation of the derivative in the definition of the Caputo fractional derivative, e.g., [25], [24], [16], [26], [37], or by the approximation of the Hadamard finite-part integral, e.g., [9], [10], [13], [14], [15], [23], [41]. Since its first appearance the L1 scheme has been extensively used in practice and currently it is one of the most popular and successful numerical methods for solving the time fractional diffusion equation.…”

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

“…The L1 scheme first appeared in the book [41] for the approximation of the Caputo fractional derivative. The L1 scheme may be obtained by direct approximation of the derivative in the definition of the Caputo fractional derivative, e.g., [28], [26], [18], [29], [47], [19], or by the approximation of the Hadamard finite-part integral, e.g., [9], [11], [15], [16], [17], [24], [51].…”

“…Hence there exists a constant C which depends only on θ and α such that, see Jin et al [21, (2.3)], (z α I + A) −1 ≤ C|z| −α , ∀ z ∈ Σ θ = {z = 0 : | arg z| < θ}. (1.3) Under the assumptions that the solutions of (1.1) are sufficiently smooth, for example u ∈ C 2 [0, T ], there are many time discretisation schemes in the literature, for example, Lubich's fractional multistep methods [7], [30], [55], [65], [61], [62], [64], the L1 scheme and its modification [28], [26], [18], [29], [47], the spectral method [58] [63], [4], [68], [57], nonpolynomial collocation method [14], discontinuous Galerkin method [39], etc.…”

Stability and Convergence Analysis of a Class of Continuous Piecewise Polynomial Approximations for Time-Fractional Differential Equations

“…Generalized Adams methods and so-called m-steps methods are utilized by [1,2]. Under the framework of product integration, recently, some new numerical approximations of the Caputo fractional derivative of order α ∈ (0, 1), named L1 method [25], L1-2 method [17], L2-1 σ method [4] and method [32], were proposed and applied for solving timefractional differential equations. These methods are based on piecewise linear or quadratic interpolating polynomials approximations.…”

“…We establish local truncation errors and global errors estimates of the numerical schemes for (1.1) in detail. In addition, we mainly study the numerical stability of the L1 method, L1-2 method, method in [32] and higher-order methods proposed in this paper. We apply the technique in [31] to the investigation of the stability regions of this type of numerical methods.…”